@article{ECP1615,
author = {Andrej Depperschmidt and Andreas Greven and Peter Pfaffelhuber},
title = {Marked metric measure spaces},
journal = {Electron. Commun. Probab.},
fjournal = {Electronic Communications in Probability},
volume = {16},
year = {2011},
keywords = {Metric measure space, Gromov metric triples, Gromov- weak topology, Prohorov metric, Population model},
abstract = {A marked metric measure space (mmm-space) is a triple $(X,r,μ)$, where $(X,r)$ is a complete and separable metric space and $μ$ is a probability measure on $X \times I$ for some Polish space $I$ of possible marks. We study the space of all (equivalence classes of) marked metric measure spaces for some fixed $I$. It arises as a state space in the construction of Markov processes which take values in random graphs, e.g. tree-valued dynamics describing randomly evolving genealogical structures in population models. We derive here the topological properties of the space of mmm-spaces needed to study convergence in distribution of random mmm-spaces. Extending the notion of the Gromov-weak topology introduced in (Greven, Pfaffelhuber and Winter, 2009), we define the marked Gromov-weak topology, which turns the set of mmm-spaces into a Polish space. We give a characterization of tightness for families of distributions of random mmm-spaces and identify a convergence determining algebra of functions, called polynomials.},
pages = {no. 17, 174-188},
issn = {1083-589X},
doi = {10.1214/ECP.v16-1615},
url = {http://ecp.ejpecp.org/article/view/1615}}